Periodic IPAD measure conjectures for 3-class tower groups

An IPAD is the invariant displayed in the source in the form of an abelianization together with the corresponding abelianizations of selected index subgroups. The notation $[3^k,3^{k+1}]$ denotes the corresponding pair of cyclic $3$-power factors, and the measure is the Cohen–Lenstra-type measure assigned to an IPAD. Periodic IPAD measure conjecture. For every integer $k\geq2$, the following four IPADs have the stated measures:

(a) $[[3,3];[3,9]^3[3^k,3^{k+1}]]$ has measure $512/3^{2k+4}$;

(b) $[[3,3];[3,3,3][3,9]^2[3^k,3^{k+1}]]$ has measure $512/3^{2k+4}$;

(c) $[[3,3];[3,3,3]^2[3^k,3^{k+1}]^2]$ has measure $2048/3^{4k+2}$;

(d) $[[3,3];[3,3,3]^2[3^k,3^{k+1}][3^{k+1},3^{k+2}]]$ has measure $512/3^{4k+2}$.

These formulas are conjectured from periodic descendant patterns and would complete the computation of measures of IPADs beginning with $[3,3]$; the supplied context gives no proof or resolution.